Approximation of Functions with Parameter-Dependent or Stochastic Shock Locations Arising from Hyperbolic Partial Differential Equations

许多科学或工程问题的数学模型都受到设计参数和随机参数的影响。一个例子是围绕翼型的流动建模。设计参数是它的形状，随机参数包括，例如，飞机穿过它之前空气中的扰动。这些参数给工程师和他们使用的数学模型提出了一些自然的问题：什么是最好的形状？我们能否计算出机翼的安全容限并限制故障概率？随着计算能力的提高，这些问题近年来引起了相当大的关注。然而，对于具有急剧转变的物理现象，只有蛮力方法可用，即使使用先进的计算机硬件也会很快变得具有挑战性。在机翼的例子中，一个这样的转变是音爆，在高速下机翼周围的压力的快速变化。该项目旨在开发新的算法，可以处理参数模型中的快速转换，比目前更有效。正在开发的方法也将适用于一系列其他工程问题，如地下水流中的环境问题或生物分子的模拟。该项目的目标是为具有参数依赖跳跃或扭结的函数开发新的近似方案，其动机是解决参数和随机双曲型偏微分方程。这些奇异性恶化了已建立的方法，如减少基方法，适当的正交分解，或多项式混沌展开的收敛速度带来了严重的挑战。最近的工作提供了一种新的方法来近似函数的跳跃不连续性和实现超多项式收敛速度的许多问题。它作为一个原则的证明，高阶方法是有利的，但它需要进一步发展，使其实用：在大多数现实的问题，跳跃不仅移动，但他们也相互作用，参数空间通常是高维的。解决这些问题是本项目的目标。新的算法将进行数字测试，特别是关于可以实现的收敛速度。此外，研究者计划证明参数函数的模型类的收敛率。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Many mathematical models of scientific or engineering problems are influenced by design parameters and random parameters. An example is the modeling of flow around an airfoil. A design parameter is its shape, and random parameters include, for example, perturbations in the air before a plane travels through it. These parameters pose some natural questions for engineers and the mathematical models they use: What is the best possible shape? Can we compute safety tolerances of the wing and limit the probability of failure? With increasing computational power, such questions attracted considerable attention in recent years. However, for physical phenomena with sharp transitions, only brute-force approaches are available, which quickly become challenging even with advanced computer hardware. In the wing example, one such transition is the sonic boom, a rapid change in the pressure around the wing at high speed. This project aims to develop new algorithms that can handle such rapid transitions in parametric models, far more efficiently than currently. The method under development will be applicable to a range of other engineering problems as well, such as environmental questions in groundwater flows or the simulation of bio-molecules.The goal of the project is the development of new approximation schemes for functions with parameter-dependent jumps or kinks, motivated by solutions of parametric and stochastic hyperbolic partial differential equations. These singularities pose serious challenges by deteriorating the convergence rates for established methods such as reduced basis methods, proper orthogonal decomposition, or polynomial chaos expansions. Recent work offers a new method to approximate functions with jump discontinuities and achieves super-polynomial convergence rates for many problems. It serves as a proof of principle that high-order methods are advantageous, but it needs to be developed further to make it practical: In most realistic problems, jumps not only move but they also interact, and parameter spaces are typically high-dimensional. Addressing these issues is the goal of this project. The new algorithms will be tested numerically, in particular regarding the convergence rates that can be achieved. In addition, the investigator plans to prove convergence rates for model classes of parametric functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.